The only record of the life and ministry of Jesus are the four Canonical Gospels: Matthew, Mark, Luke, and John. Modern historians take these narratives with a grain of salt when examining historical Jesus, as the authors of these works were clearly more interested in theology than history; they were written quite a bit after the fact; and they contradict one another, themselves and other historic documents in both matters big and small.

He offers no supporting evidence.

The gospels are Greco-Roman biographies. That's a historical genre. There are many places where the gospels' historicity can be and has been corroborated by other sources. Acts provides even more opportunity for corroboration, and it has been highly corroborated, which reflects well on the historical reliability of Luke and his gospel.

If the gospels contradict each other, as Gilad claims, that would put them in the same category as other historical sources. Critics of the infancy narratives rely heavily on Josephus in their arguments against the Biblical accounts. Josephus not only contradicts other sources, but even contradicts himself, including when he's providing autobiographical information. Furthermore, a contradiction between two gospels would only demonstrate the unreliability of one of them, not both. And why think the alleged contradictions are even relevant in this context? If two gospels contradict each other in their resurrection narratives, for example, it doesn't follow that their material on Jesus' childhood is inconsistent. Even if the gospels were contradictory on some issues related to Jesus' childhood, their high level of consistency on other matters pertaining to his childhood (the names of the parents, the premarital timing of the pregnancy, the timing of the birth, the place of birth, etc.) would have to be explained.

So, while Gilad plants some vague seeds of doubt with his comments quoted above, he doesn't substantiate any of his claims, and he ignores a lot of evidence against his position.

He goes on:

Still, these are the only documents that we, and indeed, Dionysius, have to go by.

No, we have far more than the gospels to go by. Christians passed down information from one generation to another, and so did other ancient sources (heretics, Jewish non-Christians, and Gentile non-Christians). It's highly unlikely that ancient Christians outside of the gospels and ancient non-Christians would have all, collectively, forgotten what happened and replaced a historical memory with the same unhistorical account. Thus, when the ancient extrabiblical sources are in so much agreement about when Jesus was born, his birthplace, the names of his parents, and other details, that extrabiblical evidence offers substantial corroboration of the Biblical accounts. Regarding the date of Jesus' birth in particular, since that's the focus of Gilad's article, see here.

He continues:

It seems that Matthew set Jesus’ birth during the reign of Herod to allow for his account of the Massacre of the Innocents, an infanticide allegedly ordered by Herod (2:14), but lacking any historical corroboration, nor any mention in the other gospels.

The Slaughter of the Innocents is corroborated to some extent by the Assumption Of Moses, a Jewish document likely composed shortly after the event and before the gospels, and by Macrobius, another non-Christian source, a few centuries later. And keep in mind, again, that the general acceptance of the event in extrabiblical sources, including Christian ones, can't be assumed to have no evidential value. Even if early Christian sources were just repeating what they had read in Matthew, the fact that they interpreted the account in a historical manner, didn't show any awareness of significant objections to its historicity, etc. is important. For more about the historicity of the Slaughter, see here.

We have no reason to expect any other gospel to mention the event. No other gospel narrates that portion of Jesus' life, for one thing.

Furthermore, it's commonplace for ancient sources to refer to an event that no other extant source mentions. Josephus does it often, for example. Steve Mason, a scholar who specializes in the study of Josephus, explains that Josephus is our only source for most of what he narrates (Josephus, Judea, And Christian Origins [Peabody, Massachusetts: Hendrickson Publishers, 2009], 16). Critics of the infancy narratives often tell us that Josephus discusses many misdeeds by Herod, yet doesn't mention the Slaughter. That's supposed to give us a lot of reason to doubt the historicity of Matthew's account. But the misdeeds of Herod that Josephus does mention include ones that nobody else mentions. I've read thousands of pages of material criticizing the infancy narratives. I can't remember a single critic who ever suggested that he needed corroboration from another source for Josephus' claims about Herod's misdeeds. Maybe I'm forgetting some instances in which a critic did that, but my impression is that at least the large majority of skeptics don't think such corroboration is needed for Josephus. Why do the gospels need corroboration from at least one other source, yet Josephus has no such need for corroboration?

Gilad writes:

Luke also tells us (2:1-2) that Jesus was born in Bethlehem, not in the family home in Nazareth, because of the Census of Quirinius (governor of Syria), which required Joseph to return to his ancestral home.

There is no "family home in Nazareth" in Luke's gospel. Joseph and Mary have their wedding in Bethlehem after arriving there in Luke 2. See here. Stephen Carlson published an article several years ago that corrects a lot of misconceptions about Luke's narrative, like the misconceptions Gilad repeats.

Whether the census was "of Quirinius" depends on what you mean by the term. If you want to know Luke's chronology, read 1:5. When he opens chapter 2 with a reference to "in those days" (2:1), he's referring back to the chronology already established in chapter 1. Though chronology is addressed in chapter 2, it's not his focus. Rather, he's focused on the general circumstances (not just or primarily the timing) surrounding Jesus' birth. Quirinius' governorship in 6 A.D. postdates Herod the Great (mentioned in 1:5) by about a decade. Somebody as demonstrably knowledgeable of the history he's covering as Luke was is unlikely to have thought Herod and Quirinius were in power (in the respective offices just mentioned) at the same time. And Luke's dating in 1:5 is corroborated by 3:1-2, 3:23, Matthew's gospel, and the earliest extrabiblical sources. Furthermore, as I've argued elsewhere, the contrast between the atmosphere described in Acts 5:37 and the atmosphere we see in Luke's infancy narrative suggests that Luke was aware that the events of 6 A.D. were distinct from what happened around the time of Jesus' birth.

Why does Quirinius get mentioned, then? Most likely, he's being mentioned in a parenthetical comment about the census-taking process under Augustus. The revolt that occurred in connection with the census in 6 A.D. was memorable to people interested in Israel's history, and many people in Luke's audience would have had an interest in the history of Israel. Citing Quirinius would be a simple and memorable way of identifying (and reinforcing) what census Luke had in mind. It doesn't follow that Quirinius was governor when Jesus was born. The census in verse 1 is a census-taking process that occurred over many years, in different places at different times. It was common knowledge in Luke's day that there was no one decree from Augustus that resulted in everybody being registered at the same time. Rather, there was a general policy of taking censuses across the empire, and that process could be summarized as Luke summarizes it in 2:1. (For more about Luke's language on this point, see here and here.) Quirinius wasn't governor during the entire census process. Rather, he was governor during one portion of the process that would be recognizable and memorable to Luke's audience.

Stephen Carlson has argued that verse 2 should be rendered as something like "This registration became most prominent when Quirinius was governing Syria." I don't know enough about Luke's Greek to offer much of a judgment of Carlson's translation. But I think the general thrust of his view is correct, even if he's wrong about the precise wording. Luke isn't placing Jesus' birth under Quirinius. Rather, he's making a parenthetical comment to help identify the census he's referring to, a census Quirinius presided over in part, but not in whole.

If I'm wrong, if Luke did intend to place Jesus' birth under Quirinius, then I would appeal to what I take to be the second best explanation of the passage. Luke would, in that case, likely be referring to some other governmental role Quirinius held at the time of Herod the Great. Darrell Bock discusses this view on page 908 of his Luke, Volume 1, 1:1-9:50 (Grand Rapids, Michigan: Baker Books, 1994). And see my discussion here of a passage in Justin Martyr that supports the existence of two censuses (or two phases of one census) under Quirinius. (Notice that my preferred explanation of Luke's passage allows, though it doesn't require, such a double role by Quirinius.)

If Gilad is suggesting that Luke tells us that the census required its participants to register in their place of ancestry, he's also wrong about that.

Gilad writes:

The earliest indication that Jesus was born on December 25 is a calendar from 354 C.E., more than 300 years after his time.

If early Christians knew the date of Jesus’ birth, it is somewhat strange that no record exists of such a thing before that date. It also seems rather convenient that Jesus’ birthday was December 25, which just happened to have been a festival day in the late Roman Empire for the sun god Sol Invictus.

Gerry Bowler, a historian who's studied the history of the Christmas holiday, recently told Albert Mohler that views like Gilad's are now a minority position among scholars who study the subject. Not only is there a lack of evidence that the December 25 date for Christmas was initially taken from paganism, but there's also a lack of evidence that December 25 had the sort of significance for pagans that people often claim it did.